Water Tanks, Fire Hydrants and Math

Most people looking at the water tanks atop many older Manhattan buildings might see an iconic vestige of old New York.

Glen Whitney sees a math equation.

Mr. Whitney, executive director of the Museum of Mathematics, will not have an actual museum until 2012. In the meantime, he sometimes gives walking tours to show what he describes as “this hidden collision between mathematics and the real world.”

For example, in New York City, “You always know you're close to a regular pentagon," Mr. Whitney said. The bolts that open and close fire hydrants are five-sided and not easy to turn with a typical wrench — to thwart people looking to open a hydrant on a hot summer day.

The threading is also the opposite of other bolts. "Usually it's 'righty tighty, lefty loosey,' " Dr. Whitney said. "These are what are called left-handed screws. They go the opposite way."

For the water tanks, look at the metal bands that wrap around the cylindrical walls. They are much closer together near the bottom of the tanks than at the top. Why?

The pressure of water pushing against the sides of the tank is proportional to the depth of the water. (The particulars about the size and the shape of the tank do not enter the calculations.) At twice the depth, there’s twice as much pressure, and for the water tank to hold together, the structure needs twice the strength. One way to accomplish that is to reduce by half the spacing of the bands. At three times the depth, the pressure is three times as great, and thus the spacing should be reduced to one-third the distance. The result is that the spacing gets closer together at the bottom of the tank. In mathematical terms, the spacing between bands would be described by the equation f(x) = 1/x.